Automata on Graph Alphabets

2026-02-10

AI Summary

This work extends classical finite automaton theory—traditionally confined to free strings over unstructured alphabets—to settings where the alphabet itself is a directed graph (finite or infinite), and string concatenation is constrained by vertex adjacency in the graph. By integrating algebraic language theory, graph theory, and automata models, the paper establishes foundational results for this framework: it proves adapted versions of Kleene’s theorem and the Myhill–Nerode theorem, demonstrates that the class of regular languages is not closed under complementation, and develops determinization and minimization algorithms while analyzing prefix and suffix quotient structures. The study not only constructs a comprehensive theoretical foundation for automata over graph alphabets but also outlines a pathway for generalization to more abstract presimplicial alphabets.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The theory of finite automata concerns itself with words in a free monoid together with concatenation and without further structure. There are, however, important applications which use alphabets which are structured in some sense. We introduce automata over a particular type of structured data, namely an alphabet which is given as a (finite or infinite) directed graph. This constrains concatenation: two strings may only be concatenated if the end vertex of the first is equal to the start vertex of the second. We develop the beginnings of an automata theory for languages on graph alphabets. We show that they admit a Kleene theorem, relating rational and regular languages, and a Myhill-Nerode theorem, stating that languages are regular iff they have finite prefix or, equivalently, suffix quotient. We present determinization and minimization algorithms, but we also exhibit that regular languages are not stable by complementation. Finally, we mention how these structures could be generalized to presimplicial alphabets, where languages are no more freely generated.

Problem

• graph alphabets
• structured alphabets
• finite automata
• constrained concatenation
• regular languages

Innovation

• graph alphabets
• structured automata
• Kleene theorem
• Myhill-Nerode theorem
• determinization